# Linear Systems: Continuous-Time Impulse Response Descriptions

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_188-1

## Abstract

An important input–output description of a linear continuous-time system is its impulse response, which is the response *h*(*t*, *τ*) to an impulse applied at time *τ*. In time-invariant systems that are also causal and at rest at time zero, the impulse response is *h*(*t*, 0) and its Laplace transform is the transfer function of the system. Expressions for *h*(*t*, *τ*) when the system is described by state-variable equations are also derived.

## Keywords

Linear systems Continuous-time Time-invariant Time-varying Impulse response descriptions Transfer function descriptions## Introduction

*H*(

*t*,

*τ*) is the

*impulse response matrix*of the system (1). To explain, consider first a single-input single-output system:

*f*(

*t*),

*y*

_{ I }(

*t*) is

*t*when an impulse is applied at the input at time \(\hat{t}\). So in (4),

*h*(

*t*,

*τ*) is the response at time

*t*to an impulse applied at time

*τ*. Clearly if the

*impulse responseh*(

*t*,

*τ*) is known, the response to any input

*u*(

*t*) can be derived via (4), and so

*h*(

*t*,

*τ*) is an input/output description of the system.

*u*(

*τ*) in (1) be zero except the

*j*th component, then

*h*

_{ ij }(

*t*,

*τ*) denotes the response of the

*i*th component of the output of system (1) at time

*t*due to an impulse applied to the

*j*th component of the input at time

*τ*with all remaining components of the input being zero.

*H*(

*t*,

*τ*) = [

*h*

_{ ij }(

*t*,

*τ*)] is called the

*impulse response matrix*of the system.

*t*=

*t*

_{0}(i.e., if

*u*(

*t*) = 0 for

*t*≥

*t*

_{0}, then

*y*(

*t*) = 0 for

*t*≥

*t*

_{0}),

*y*(

*t*

_{0}) = 0 and (9) becomes

*H*(

*t*−

*τ*)) since only the elapsed time (

*t*−

*τ*) from the application of the impulse is important. Then (10) becomes

*t*

_{0}= 0 without loss of generality. Equation (11) is the description for causal, time-invariant systems, at rest at

*t*= 0.

*y*(

*t*),

*u*(

*t*) and \(\hat{H}(s)\) is the Laplace transform of the impulse response

*H*(

*t*). \(\hat{H}(s)\) is the

*transfer function matrix*of the system. Note that the transfer function of a linear, time-invariant system is typically defined as the rational matrix \(\hat{H}(s)\) that satisfies (12) for any input and its corresponding output assuming zero initial conditions, which is of course consistent with the above analysis.

## Connection to State-Variable Descriptions

*x*(

*t*

_{0}) = 0, i.e., the system is at rest at

*t*

_{0}. Here \(\Phi (t,\tau )\) is the state transition matrix of the system defined by the Peano-Baker series:

*x*(

*t*

_{0}) = 0.

*τ*= 0)

*x*(0) = 0) and taking Laplace transform of both sides.

Finally, it is easy to show that equivalent state-variable descriptions give rise to the same impulse responses.

## Summary

The continuous-time impulse response is an external, input–output description of linear, continuous-time systems. When the system is time-invariant, the Laplace transform of the impulse response *h*(*t*, 0) (which is the output response at time *t* due to an impulse applied at time zero with initial conditions taken to be zero) is the transfer function of the system – another very common input–output description. The relationships with the state-variable descriptions are shown.

## Cross-References

“Linear Systems: Continuous-Time, Time-Invariant, State Variable Descriptions,” Panos J. Antsaklis

“Linear Systems: Continuous-Time, Time-Varying, State Variable Descriptions,” Panos J. Antsaklis

## Recommended Reading

External or input–output descriptions such as the impulse response and the transfer function (in the time-invariant case) are described in several textbooks below.

## Bibliography

- Antsaklis PJ, Michel AN (2006) Linear systems. Birkhauser, BostonGoogle Scholar
- DeCarlo RA (1989) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
- Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
- Rugh WJ (1996) Linear systems theory, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
- Sontag ED (1990) Mathematical control theory: deterministic finite dimensional systems. Texts in applied mathematics, vol 6. Springer, New YorkGoogle Scholar